A Definition for Large Eddy Simulation Approximations of the Navier-Stokes Equations

A DEFINITION FOR LARGE EDDY SIMULATION APPROXIMATIONS OF THE NAVIER-STOKES EQUATIONS J.-L. GUERMOND1,3 AND S. PRUDHOMME2 Abstract. This paper proposes a mathematical definition of Large Eddy Sim- ulation (LES) for three-dimensional turbulent incompressible viscous flows. We propose to call LES approximation of the Navier–Stokes equations any sequence of finite-dimensional approximations of the velocity and pressure fields that converge (possibly up to subsequences) in the Leray class to a suitable weak solution. 1. Introduction 1.1. Introductory comments. At the present time, computer predictions of turbulence phenomena by so-called Direct Numerical Simulation (DNS) of the Navier- Stokes equations are still considered a formidable task for Reynolds numbers larger than a few thousands. Since the times of Boussinesq and Reynolds, numerous turbulence models based on time-averaged or space-averaged quantities (Reynolds Averaged Navier–Stokes models, k- models, etc.) have been developed and then used in engineering applications as a means of overcoming, though often with lim- ited success, the lack of sufficient computer resources required by DNS. During the past forty years a new class of turbulence models, collectively known as Large Eddy Simulation (LES) models [23], has emerged in the literature. These models are founded on the observation that representing the whole range of flow scales may not be important in many engineering applications as one is generally inter- ested only in the large scale features of the flow. With this objective in mind, LES modelers have devised artifacts for representing the interactions between the unreachable small scales and the large ones. An extended variety of LES models is now available; see e.g. [6, 12, 18] for reviews. However no satisfactory mathematical theory for LES has yet been proposed (for preliminary attempts of formalization see [15, 19, 18]). More surprisingly, no mathematical definition of LES has been stated either (to the best of our knowledge), although some qualitative attempts have been made in this direction. For instance the following formal definition is proposed in [8]: “We define a large eddy simulation as any simulation of a turbulent flow in which the large-scale motions are explicitly resolved while the small-scale motions are represented approximately by a model (in engineering nomenclature) or parameterization (in the geosciences).” The objective of this paper is to go beyond qualitative statements like the one above by proposing a mathematical definition of LES approximations of the Navier–Stokes equations. Date: Draft version: May 6, 2004. 1991 Mathematics Subject Classification. 35Q30, 65N35, 76M05. Key words and phrases. Navier–Stokes equations, Turbulence, Large Eddy simulation. 1 2 J.-L. GUERMOND AND S. PRUDHOMME This paper is organized as follows: in the remainder of the Introduction, we recall the notion of suitable weak solutions of the Navier-Stokes equations as it will be included into our definition of LES approximations. We state in Section 2 the definition that we propose for LES approximations and discuss some of its features. In particular, we introduce the concept of pre-LES-models. In Section 3, we list existing models that fall into the category of the pre-LES-models. We proceed in Section 4 by providing examples of LES approximations and for each of these examples we show how to use our definition of LES to determine the relationship between the discretization parameter and the cutoff scale parameter. Concluding remarks are finally reported in Section 5. 1.2. Suitable weak solutions. It is generally accepted that the Navier–Stokes equations stand as a reasonable model to predict the behavior of turbulent incompressible flows of viscous fluids. Upon denoting by Ω ⊂ R3 the open smooth connected domain occupied by the fluid, ]0,T [ some time interval, u the velocity field, and p the pressure, the problem is formulated as follows: 2 ∂tu + u·∇u + ∇p − ν∇ u = f in QT , ∇·u = 0 in QT , (1.1) u| = 0 or u is periodic, Γ u|t=0 = u0, where QT = Ω × (0,T ), Γ is the boundary of Ω, u0 the solenoidal initial data, f a source term, ν the viscosity, and the density is chosen equal to unity. Henceforth, we assume that (1.1) has been nondimensionalized, i.e. ν is the inverse of the Reynolds number. To implicitly account for boundary conditions, we introduce ( H1(Ω) If Dirichlet conditions, (1.2) X = 0 {v ∈ H1(Ω), v periodic} If periodic conditions. (1.3) V = {v ∈ X, ∇·v = 0}, L2 (1.4) H = V In mathematical terms, the turbulence question is an elusive one. Since the bold definition of turbulence by Leray in the 1930’s [24], calling solution turbulente any weak solution of the Navier–Stokes equations, progress has been frustratingly slow. The major obstacle in analyzing the Navier–Stokes equations has to do with the question of uniqueness of solutions in three dimensions, a question not yet solved owing to the possibility that the occurrence of so-called vorticity bursts reaching scales smaller than the Kolmogorov scale cannot be excluded. If weak solutions are not unique, a fundamental question is then to distinguish the physically relevant solutions. A possible piece of the maze might be the notion of suitable weak solutions proposed by Scheffer [30]: Definition 1.1 (Scheffer). A weak solution to the Navier-Stokes equation (u, p) 2 ∞ 2 3 is suitable if u ∈ L (0,T ; X) ∩ L (0,T ; L (Ω)), p ∈ L 2 (QT ) and the local energy balance 1 1 1 (1.5) ∂ ( u2) + ∇·(( u2 + p)u) − ν∇2( u2) + ν(∇u)2 − f · u ≤ 0 t 2 2 2 is satisfied in the distributional sense. A DEFINITION OF LES 3 It is remarkable that, to date, it is for the class of suitable weak solutions that the best partial regularity result has been proved, as stated in the so-called Caffarelli- Kohn-Nirenberg (CKN) Theorem, see Caffarelli et al. [2], Lin [25], Scheffer [30]. This theorem states that the one-dimensional Hausdorff measure of singular points of suitable solutions is zero. In other words, suitable weak solutions are almost classical (i.e., smooth). Whether these solutions are indeed classical is still far from being clear (see e.g. Scheffer [31]). Nevertheless, the result of the CKN Theorem is our principal motivation for incorporating the notion of suitable weak solutions in the definition of LES models. Note that, by analogy with nonlinear conservative laws, the condition (1.5) can be viewed as an entropy-like condition in the sense that it is expected to select physical solutions of (1.1). Duchon and Robert [7] have given an explicit form of the distribution D(u) that is missing in the left-hand side of (1.5) to reach equality. For a smooth flow, the distribution D(u) is zero; but for nonregular flow D(u) may be nontrivial. Suitable solutions are those which satisfy D(u) ≥ 0, i.e., if singularities appear, only those that dissipate energy are admissible. 2. The main definition We believe that Large Eddy Simulation approximations should be defined as solutions of finite-dimensional problems which can be implemented on digital comput- ers. In this respect, a reasonable definition should fill the gap between the current so-called LES modeling theories and their various numerical implementations. The existence of this gap has been repeatedly acknowledged in the literature without being seriously addressed; see e.g. Ferziger [9]: “In general, there is a close connection between the numerical methods and the modeling approach used in simulation; this connection has not been sufficiently appreciated by many authors.” Finally, LES approximations should select physical solutions of the Navier-Stokes equations under the appropriate limiting process. Hence we propose the following definition for LES: ∞ 2 2 Definition 2.1. A sequence (uγ , pγ )γ>0 with uγ ∈ L (0,T ; L (Ω)) ∩ L (0,T ; X) 0 2 and pγ ∈ D (]0,T [,L (Ω)) is said to be a LES approximation to (1.1) if 2 (i) There are two finite-dimensional vectors spaces Xγ ⊂ X and Mγ ⊂ L (Ω) 1 0 such that uγ ∈ C ([0,T ]; Xγ ) and pγ ∈ C ([0,T ]; Mγ ) for all T > 0. (ii) The sequence converges (up to subsequences) to a weak solution of (1.1), 2 0 2 say uγ * u weakly in L (0,T ; X) and pγ → p in D (]0,T [,L (Ω)). (iii) The weak solution (u, p) is suitable. At this point, we want to emphasize that two parameters are actually hidden in the above definition. Since Xγ and Mγ are finite-dimensional, there is a discretization parameter h associated with the size of the smallest scale that can be 3 represented in Xγ , roughly dim(Xγ ) = O((L/h) ) where L = diam(Ω). In the following we also consider a regularization parameter ε associated with some regularization of the Navier–Stokes equations. This parameter is the lengthscale of the smallest eddies that are allowed to be nonlinearly active in the flow (large eddies). In the above definition, the parameter γ is yet an unspecified combination of the two parameters h and ε. In practice, the construction of a LES model will be decomposed into the following three steps: 4 J.-L. GUERMOND AND S. PRUDHOMME (1) Construction of what we hereafter call the pre-LES-model. This step con- sists of regularizing the Navier-Stokes equations by introducing the parameter ε. The purpose of the regularization technique is to yield a well-posed problem for all times. Moreover, the limit solution of the pre-LES-model must be a weak solution to the Navier–Stokes equations as ε → 0 and should be suitable.

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